Let \(w\)  be any primitive prime divisor \(w\)  of \(q^{e}-1\)  dividing \(k+1\)  and
let \(W\)  be a Sylow \(w\) -subgroup of \(G_{x}\) . Then \(w\equiv 1 \pmod {ef}\)  and
hence \((u,ef)=1\) . Thus \(w\)  is coprime to \(\left|\mathrm {Out(}X\mathrm {)}\right|\)  (see [1]}, Table 5.1.A). Therefore, \(W\leqslant X_{x}\) .
Assume that \(W\)  fixes a further point \(y\)  of \(\mathcal {D}\) . Then \(\left|y^{G_{x}}\right|\)  has order prime to \(w\) . However, this is impossible,
since \(k+1\)  divides \(\left|y^{G_{x}}\right|\)  by Lemma REF (3), and since \(u \mid k+1\) .
Thus \(x\)  is the unique point fixed by \(W\) . Then \(N_{X}(W)\leqslant N_{G}(W)\leqslant G_{x}\) , and hence \(N_{X}(W)\leqslant X_{x}\) , as \(W\leqslant X_{x}\) .
The group \(W\)  preserves a unique \((e-1)\) -subspace \(\mathcal {H}\)  of \(PG_{n-1}(q)\)  and acts
faithfully and irreducibly on it (see [2]}, Theorem 3.5). So does \(N_{X}(W)\) . It is not difficult to
see that \(W\)  lies in a cyclic subgroup \(Q\)  of \(X_{\mathcal {H}}\)  inducing a
Singer cyclic subgroup \(\bar{Q}\)  on \(\mathcal {H}\) . The order of \(\bar{Q}\)  is
well known and, for instance, can be found in [3]},[4]},[5]}, or determined in [1]}, Section 4.3. Thus \(\Phi _{ef}^{\ast }(p)\)  divides the order of \(\bar{Q}\) , and
hence that of \(Q\) . Then \(\Phi _{ef}^{\ast }(p)\)  divides \(\left|N_{X}(W)\right|\) , and hence \(\left|X_{x}\right|\) , and as \(Q\leqslant N_{X}(W)\) . Finally, if \(e=n\) , clearly \(\mathcal {H}=PG_{n-1}(q)\)  and hence \(Q\)  is a Singer cyclic subgroup of \(X\) .
